The vanishing-power conjecture for smooth functions on projective space

Let $k$ and $\ell$ be positive integers. Vanishing-power conjecture. There exists a smooth nonconstant function $v$ on $\mathbb{P}^k$ such that

$$(dd^c v)^\ell\equiv 0$$

if and only if

$$\ell>\left[\frac{k+2}{2}\right].$$

The proposed threshold is motivated by the claim that $K^+(v)$ and $K^+(-v)$ are $(\ell-1)$-pseudoconcave and should intersect when $2(\ell-1)\leq k$; the source gives no evidence that the assertion has been proved or disproved.